Search for lepton flavour violation in $\tau$ decays at the LHCb experiment

Decays violating the charged lepton flavour number are allowed in the Standard Model only through neutrino oscillations, with branching fractions suppressed to unmeasurable levels. Any observation of such transitions would unambiguously indicate the existence of phenomena beyond the Standard Model. The LHCb Collaboration published the first result on the $\tau^- \rightarrow \mu^+ \mu^- \mu^-$ decay mode from a hadron collider. The limit $B(\tau^- \rightarrow \mu^+ \mu^- \mu^-) < 8.0 \cdot 10^{-8}$ is set at $90\%$ C.L. with data collected in 2011 at LHC running at a center-of-mass energy of $\sqrt{s} = 7 ~\mathrm{TeV}$ and corresponding to an integrated luminosity of $1.0 ~\mathrm{fb}^{-1}$. The rare nature of such a decay makes the discrimination between signal and background fundamental for the analysis. Geometrical properties of the reconstructed $\tau$ decay play an important role in background rejection. One of the most important geometrical variable is the track isolation, which is a measure for the number of background tracks that form vertexes with the signal candidate tracks. Procedures for the optimization of the track isolation variable have been studied in order to enhance the background rejection power. The study presented in this thesis contributes to the analysis on the $\tau^- \rightarrow \mu^+ \mu^- \mu^-$ decay mode performed by the LHCb Collaboration with data collected during 2012 $\sqrt{s} = 8 ~\mathrm{TeV}$ and corresponding to an integrated luminosity of $2.1 ~\mathrm{fb}^{-1}$.